L9a6

L9a5

L9a7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a6 at Knotilus!
	L9a6 is $9_{29}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a6's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X18,13,5,14 X12,17,13,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 7, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uv^{4}+3uv^{3}-3uv^{2}+2uv-u-v^{5}+2v^{4}-3v^{3}+3v^{2}-2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {7}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {6}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+3a^{9}z^{3}+8a^{9}z+5a^{9}z^{-1}-2a^{7}z^{5}-7a^{7}z^{3}-7a^{7}z-3a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-2a^{5}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-2z^{5}a^{13}+2z^{3}a^{13}-3z^{6}a^{12}+3z^{4}a^{12}-z^{2}a^{12}-3z^{7}a^{11}+4z^{5}a^{11}-6z^{3}a^{11}+6za^{11}-2a^{11}z^{-1}-z^{8}a^{10}-5z^{6}a^{10}+14z^{4}a^{10}-15z^{2}a^{10}+5a^{10}-6z^{7}a^{9}+15z^{5}a^{9}-21z^{3}a^{9}+17za^{9}-5a^{9}z^{-1}-z^{8}a^{8}-4z^{6}a^{8}+14z^{4}a^{8}-13z^{2}a^{8}+5a^{8}-3z^{7}a^{7}+8z^{5}a^{7}-10z^{3}a^{7}+9za^{7}-3a^{7}z^{-1}-2z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-2za^{5}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 2 1 -1 -8               3     3 -10             3 2     -1 -12           5 3       2 -14         3 4         1 -16       4 4           0 -18     1 3             2 -20   1 4               -3 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L9a5 L9a7